The length of the shortest closed geodesic in a closed Riemannian $3$-manifold with nonnegative Ricci curvature
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- by Ezequiel Barbosa and Yong Wei PDF
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Abstract:
In this note we discuss the problem of finding an upper bound on the length of the shortest closed geodesic in a closed Riemannian 3-manifold in terms of the volume. More precisely, we show that there exists a positive universal constant $C$ such that, for every Riemannian 3-manifold $(M^3,g)$ with $Ric_g\geq 0$, at least one of the following assertions holds: (i). $Sys_g(M)\leq C Vol_g(M)^{\frac 13}$, where $Sys_g(M)$ denotes the length of the shortest closed geodesic in $M^3$; (ii). $M^3$ is diffeomorphic to $\mathbb {S}^3$ and there exists a closed minimal surface $\Sigma _0$ embedded in $M^3$, with index 1, and $A_g(\Sigma _0)\leq C Vol_g(M)^{\frac {2}{3}}$. This gives a partial answer to the problem proposed in Gromov’s paper written in 1983.References
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Additional Information
- Ezequiel Barbosa
- Affiliation: Departamento de Matemática, Universidade Federal de Minas Gerais (UFMG), Caixa Postal 702, 30123-970, Belo Horizonte, MG, Brazil
- Email: ezequiel@mat.ufmg.br
- Yong Wei
- Affiliation: Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, United Kingdom
- MR Author ID: 1036099
- ORCID: 0000-0002-9460-9217
- Email: yong.wei@ucl.ac.uk
- Received by editor(s): November 7, 2014
- Received by editor(s) in revised form: June 9, 2015, and October 30, 2015
- Published electronically: March 16, 2016
- Communicated by: Michael Wolf
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 4001-4007
- MSC (2010): Primary 53C42, 53C22
- DOI: https://doi.org/10.1090/proc/13042
- MathSciNet review: 3513555